AN IRISH TRAGEDY exists in F an element y such that ay = 1 (whence ya = 1, by I)-1 From these simple postulates the whole of common algebra follows. A word or two about some of the statements may be helpful to those who have not seen algebra for years. In II, the statement (a — b) -f c = a + (b 4- c), called the associative law of addition, says that if a and b are added, and to this sum is added c, the result is the same as if a and the sum of b and c are added. Similarly with respect to multiplication, for the second statement in II. The third statement in II is called the distributive law. In III a 'zero' and 'unity' are postulated; in IV, the postulated x gives the negative of a; and the first paren- thetical remark in V forbids 'division by zero". The demands in Postulate I are called the commutative laws of addition and m ultiplication respectively. Such a set of postulates may be regarded as a distillation of experience. Centuries of working with numbers and getting useful results according to the rules of arithmetic - empirically arrived at - suggested most of the rules embodied in these precise postulates, but once the suggestions of experience are understood, the interpretation (here common arithmetic) fur- nished by experience is deliberately suppressed or forgotten, and the system defined by the postulates is developed abstractly* on its own merits, by common logic plus mathematical tact. Notice in particular IV, which postulates the existence of negatives. We do not attempt to deduce the existence of nega- tives from the behaviour of positives. When negative numbers first appeared in experience, as in debits instead of credits, they, as numbers, were held in the same abhorrence as 'unnatural" monstrosities as were later the 'imaginary' numbers V — I, V — 2, etc., arising from the formal solution of equations such as xz -f 1 = 0, x* -f 2 = 0, etc. If the reader will glance back at what Gauss did for complex numbers he will appreciate more fully the complete simplicity of the following partial statement of Hamilton's original way of stripping 'imaginaries' of their silly, purely imaginary mystery. This simple tiling was one of the steps which led Hamilton to his quaternions, although strictly it has nothing to do with them. It is the method and the 391